Geophys. J . In(. (1992) 109, 162-170
Correspondence between theory and observations of polar motion
Richard S. Gross
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91 109 USA
Accepted 1991 November 4. Received 1991 November 4; in original form 1991 March 29
SUMMARY
The Earth’s orientation in space changes in response to the action of a variety of
torques, generated both externally and internally. External torques arising from the
gravitational forces of the Sun and Moon act upon the Earth, causing it to undergo
periodic motions known as the lunisolar precession and nutations. Internal
dynamical processes that change the deformable Earth’s inertia tensor, or that
generate relative angular momentum, also cause the Earth’s orientation in space to
change, the resulting motions being known as wobble (or polar motion) and changes
to the length-of-day. Historically, it was thought that observations determined the
location of the rotation pole within some rotating reference frame fixed to the solid
Earth, and theoretical expressions for the nutations and wobble were developed and
given in terms of the Earth’s rotation axis. However, it has been (relatively recently)
argued that observatories, being located on the surface of the Earth, are more
nearly moving with the Earth’s surface, and hence observations more nearly reflect
the motion of the Earth’s figure axis, rather than its rotation or angular momentum
axes. This argument was accepted by the International Astronomical Union, and the
current 1980 Theory of Nutation refers to an axis, the celestial ephemeris pole, that
is more closely associated with the Earth’s figure axis than it is with the Earth’s
rotation axis. Polar motion values, as reported by modern Earth rotation services,
give the location of the celestial ephemeris pole within some rotating, body-fixed
reference frame. The celestial ephemeris pole does not correspond to either the
Earth’s instantaneous figure axis, its instantaneous rotation axis, or its instantaneous
angular momentum axis, but rather corresponds to an axis that exhibits no nearly
diurnal motions in either the terrestrial, body-fixed reference frame or the celestial,
space-fixed frame.
The focus of this paper is not on the nutational motions of the Earth generated by
external torques, but rather on the wobble. The goal of this paper is to write in
terms of reported values the standard theoretical equation describing the Earth’s
wobble, namely, the linearized conservation of angular momentum equation known
as the Liouville equation. In terms of the location m ( t ) = r n , ( t ) + i r n 2 ( t ) of the
rotation pole, the Liouville equation is usually written as
m
i dm
+ -=~
CJO dt
i d
( t-) - - x ( t )
!2 dt
where the complex-valued X-functions are functions of perturbations to the Earth’s
inertia tensor and relative angular momentum. In order to rewrite this equation in
terms of the location p ( t ) = p l ( t ) ipz(t) of the reported pole, a relation between
the locations of the rotation pole and the celestial ephemeris pole must be obtained.
This is accomplished in the time domain by considering the properties of the
time-dependent transformation matrix that relates components of a position vector
in the rotating, body-fixed frame to its components in the celestial, space-tixed
frame, the resulting relation being
+
i dp
m ( t ) = p ( t )- --.
52 dt
162
Theory and observations of polar motion
163
Using this expression relating the location of t h e rotation pole to that of t h e celestial
ephemeris pole, the linearized conservation of angular momentum equation
becomes
Thus, there is no 2-term in t h e Liouville equation describing long-period polar
motions when it is written in terms of reported valves.
Key words: E a r t h rotation, length-of-day, polar motion, wobble.
INTRODUCTION
Conceptually, the Earth’s rotation is easily understood.
External torques operate on the Earth, affecting the Earth’s
angular momentum vector, causing it to change in space.
External is taken here to mean everywhere external to the
surface of the Earth, with the Earth’s atmosphere and
hydrosphere being considered to be internal to the Earth’s
surface. By definition, the angular momentum vector is the
inner product of the Earth’s inertia tensor with its rotation
vector. Thus, changes of the Earth’s inertia tensor and
rotation vector will be associated with changes of the
angular momentum vector. The resulting (time-dependent)
changes of the Earth’s orientation in space generated by
these external torques are known as the luni-solar
precession and nutations. Internal processes can also change
the Earth’s inertia tensor, or generate relative angular
momentum, thereby changing the Earth’s rotation. However, in the case of internal processes, the Earth’s total
angular momentum vector is conserved and does not change
in space. Complications arise in practice due to the Earth’s
ability to deform in response to forces of both external and
internal origin, causing the Earth’s inertia tensor to change,
thereby also changing the Earth’s rotational motion.
In the past, there has been some confusion concerning the
observations of the rotational motions of the Earth, much of
it centred on understanding what is actually being
determined by these observations. Is it the location of the
Earth’s rotation axis, angular momentum axis, figure axis,
or some other axis? Historically, it was thought that
observations determine the location of the Earth’s rotation
axis. Theories of the Earth’s rotation, encompassing both
the nutations generated by the gravitational forces of
heavenly bodies external to the Earth (e.g. Woolard 1953),
and the wobble and changes in the length-of-day generated
by forces internal to the Earth (e.g. Munk & MacDonald
1960), were developed and expressed in terms of the Earth’s
rotation axis. It was only relatively recently that Jeffreys
(1959, 1963) and, later, Atkinson (1973) argued that
observatories, being located on the surface of the Earth, are
moving in space more nearly with the Earth’s figure axis,
rather than its rotation or angular momentum axes, and
that, therefore, a theory of nutation should refer to the
figure axis. This point of view was accepted by the
International Astronomical Union (IAU), and their 1980
Theory of Nutation refers to an axis associated with the
Earth’s figure axis (Seidelman 1982).
The subject of this paper, however, is not the nutational
motion of the Earth generated by external torques, but
rather the rotational motion of the Earth known as the
wobble and changes in the length-of-day that are generated
by internal dynamical processes. The theory of the wobble is
currently written in terms of the location of the rotation
axis. However, Earth rotation services do not report the
location of the Earth’s rotation axis, but rather report the
location of the celestial ephemeris pole, which is an axis
more closely associated with the Earth’s figure axis than it is
with the Earth’s rotation axis.
Classical astronomic latitude-longitude observations, as
well as observations by the modern space-geodetic
techniques of very long baseline interferometry (VLBI),
satellite laser ranging (SLR) and lunar laser ranging (LLR),
can determine the orientation in space of a terrestrial
reference frame defined by the locations of the observatories
(see below). That is, Earth rotation observations can
determine the orientation of a rotating, body-fixed reference
frame (defined by the location of a set of observatories) with
respect to a celestial, space-fixed reference frame (defined
by the location of a set of stars or radio sources, or defined
by the orbital motion of the Moon or some artificial
satellite). The goal of this paper is to write the theory of the
Earth’s rotation (describing the Earth’s wobble and changes
in the length-of-day) in terms of reported values. It will be
shown that the difference between the linearized conservation of angular momentum equation written in terms of the
location of the rotation axis (which is its usual form) and the
Liouville equation derived in this paper (written in terms of
the location of the celestial ephemeris pole) is only
important for the wobble, with no modifications being
needed to the equation describing changes in the rotation
rate (or length-of-day) of the Earth.
Observations can determine the Earth’s orientation in
space, but cannot distinguish between an external or
internal origin for some observed change in the Earth’s
orientation. For simplicity, it will be assumed throughout
this paper that no external torques are acting upon the
Earth model. In this case, changes in the Earth’s rotation
are due solely to internal dynamical processes, the resulting
motions being known as wobble, or polar motion, and
changes in the length-of-day.
THEORY OF THE EARTH’S ROTATION
Consider some deformable Earth model that is rotating with
respect to an inertial reference frame (the i-frame). Let this
Earth model be observed from a reference frame that is
itself rotating with angular velocity w with respect to the
inertial reference frame. This rotating reference frame will
later be attached to the Earth model in some prescribed
manner, but for now consider w to be completely arbitrary.
R . S. Gross
164
Conservation of angular momentum within the rotating
reference frame is expressed by:
their time derivatives (Wahr 1982):
dL
=Q2(C - A )
-twXL=t
(Q2 Af,,
+ 9 A h , + Q-
dt
at
(9)
where the time derivative denotes change with respect to the
rotating reference frame, and t represents external torques
acting o n the Earth model. T h e angular momentum L can in
general b e separated into two terms as
+
L=I.w h
(2)
where I is the inertia tensor, and h is the relative angular
momentum due t o motion of the Earth model relative to the
rotating reference frame.
Initially, let the Earth model be in a state of uniform
rotation at the rate 51 about its figure axis which coincides
with the &-axis of the inertial reference frame (the gi-axis).
Let the rotating reference frame be attached to the Earth
model such that it is also initially rotating at the rate Q
about its %-axis which coincides with the i$-axis. Furthermore, let the rotating reference frame be oriented within the
Earth model such that the inertia tensor of the Earth model
is diagonal within the rotating reference frame. Finally, let
no external torque t or relative angular momentum h be
present:
where C,,, is the greatest principal moment of inertia of the
mantle and the factor 1.61 accounts for the nonparticipation of the core in the wobble and the assumption
that the solid Earth and oceans respond linearly to changes
in rotation. By defining the complex quantities
m
= m 1 + im,,
Ah = A h ,
q = ql + iy,,
+ i Ah,,
Al= At,,
(12)
+ i At,,,
equations ( 6 ) and (7) can b e written more compactly as
i
m+-m=q
0,
where by (9) and (10):
'
= Q2(C- A )
where t h e subscript o denotes initial quantities. This initial
state is perturbed by some internal geophysical event
causing (in general time dependent) perturbations A1 to the
inertia tensor and Ah to the relative angular momentum,
which, by conservation of angular momentum, induces
perturbations Am to the angular velocity. The perturbations
to the angular velocity a r e usually written in terms of the
dimensionless quantities mi defined by
Am = Q ( m , , m,, m,)*.
(4)
Assuming that all perturbations are small,
Im,J << 1,
1AJJ << C,
JAhil<< QC,
(5)
so that their products can be neglected, it can be shown
(e.g. Munk & MacDonald 1960; Lambeck 1980; Wahr 1982;
Barnes et al. 1983) that the linearized conservation of
angular momentum equation is
1
ml--m2=vl,
00
1
m,+-m,
0"
= v3,
+ Q A h - iQ
__
d'
dt
- i-d A h )
dt
.
(14)
The total rotation vector is
w = W,
+ AW
(15)
which by (3) and (4) has components in the rotating
reference frame:
+
w = Q ( m , , m,, 1 m3)T.
( 16)
Since, to first order in the m i , the magnitude lo(of the
rotation vector is Q(l + m,), then m i , m,, 1 are the
direction cosines of the rotation vector with respect to the
coordinate axes of the rotating reference frame. Alternatively, and also t o first order in the mi, m l and m , give the
angular offset of the rotation vector from the %-axisof the
rotating reference frame in the t- and ?-directions,
respectively, of the rotating reference frame.
If the Earth model were rigid, then all its mass elements
would have the same angular velocity with respect to the
inertial reference frame, and the rotating reference frame
could simply be chosen t o rotate with this angular velocity.
In this case, the velocity in space of some mass element of
the Earth model located at r is simply given by
dr _
-oXr.
= q2,
(7)
(8)
where the dot denotes time differentiation, and uo is the
complex-valued frequency of the Chandler wobble. T h e vi
are known as excitation functions which, in the absence of
external torques, are functions of the perturbations t o the
inertia tensor and the relative angular momentum as well as
m3
(SZ2 A 1
dt
However, since the Earth model is deformable, different
mass elements may, in general, rotate with different angular
velocities. In this case, the rotating reference frame can be
chosen to rotate with some mean angular velocity ( w )
defined such that the quantity
Theory and observations of polar motion
is a minimum where p is the density of the mass element
located at r and the volume integral extends over that part
of the Earth model of interest. If this volume integral
extends over the crust and mantle of the Earth model, then
the rotation vector thus defined determines the ‘Tisserand
mean mantle’ rotation vector (e.g. Munk & MacDonald
1960, p. lo), and it can be shown (e.g. Jeffreys 1976, p. 293)
that within a reference frame rotating with the ‘Tisserand
mean mantle’, the relative angular momentum of the crust
and mantle is zero. Wahr (1981) has generalized the concept
of the ‘Tisserand mean mantle’ to that of the ‘Tisserand
mean surface’ by replacing the volume integral in (18) with a
surface integral extending over the surface of the Earth
model. Since the stations that observe the Earth’s rotational
properties are located on the surface of the solid Earth,
observations of the Earth’s rotational properties most nearly
reflect the rotation of the ‘Tisserand mean surface’. The
rotating reference frame defined herein is chosen to rotate
with the ‘Tisserand mean surface’, and is thereby considered
to be ‘fixed’ to the Earth model.
At this point the motion of the rotating, body-fixed
reference frame has been specified. It now remains to
specify its orientation within the Earth model. The origin of
the rotating, body-fixed reference frame is first placed at the
centre-of-mass of the Earth model. The terrestrial frame, or
t-frame, is then defined by aligning its coordinate axes with
the axes of some chosen conventional terrestrial reference
frame (CTRF). A CTRF is defined in practice by specifying
the coordinates of a set of observatories located on the
surface of the Earth. Examples of conventional terrestrial
reference frames are the Conventional International Origin
(CIO; e.g. Lambeck 1988, p. 27) and the various solutions
of the International Earth Rotation Service (IERS)
Terrestrial Reference Frame such as ITRF89 (see the IERS
Annual Report for 1989). The results of Earth rotation
observations are usually given within some CTRF.
O B S E R V A T I O N S OF T H E E A R T H ’ S
ROTATION
The theory of the Earth’s rotation, as outlined above,
relates changes of the rotation vector to the geophysical
processes causing these changes. Changes of the rotation
vector are specified by the values of the mi which locate the
rotation axis within a rotating, body-fixed reference frame.
If equations (6-8) are to be used to infer properties of the
Earth from Earth rotation observations, then these
observations must yield the time-dependent location of the
rotation axis within this rotating, body-fixed reference
frame. Historically, this was thought to be the case, and for
this reason Woolard (1953) developed a series of coefficients
describing the nutational motion of the rotation axis of a
rigid Earth model. These coefficients determined by
Woolard (1953) were then adopted by the International
Astronomical Union (IAU) as part of their System of
Astronomical Constants for use starting in 1960. However,
currently operating Earth rotation services do not report the
location of the rotation pole within some rotating,
body-fixed reference frame, but rather report the location of
a pole, known as the celestial ephemeris pole, that is more
closely associated with the Earth’s figure axis than it is with
the Earth’s rotation axis.
165
Jeffreys (1959, 1963) and Atkinson (1973) argued that
since observatories are attached to the surface of the solid
Earth, and that since it is the nutational motion of the
observatories that is actually observed, then for theory to
correspond to observations, the theoretical series of
nutational coefficients should be given for the figure axis of
some Earth model. In the case of a deformable Earth model
care must be taken in specifying this figure axis, but they
argued that conceptually the figure axis should be used since
this is the axis (rather than the rotation or angular
momentum axes) that is most firmly ‘tied’ to the locations of
the observatories. This point of view was accepted by the
IAU when recommending that the series of nutation
coefficients for the B-axis of Wahr (1981) be adopted as the
1980 IAU Theory of Nutation. Brief histories of these
events have been given by Kinoshita et al. (1979) and
Seidelmann (1982).
Atkinson (1975) explicitly showed that meridian observations of stellar declinations involve the Earth’s axis of figure,
concluding that nutations should be computed for this axis,
rather than the axis of rotation. Ooe & Sasao (1974) have
explicitly shown that astronomical latitude observations at a
single station, involving measurements of the zenith
distances of stars at their meridian passage [see, e.g., Munk
& MacDonald (1960) for a review of the optical astrometric
observing techniques], determines the orientation of the
local vertical in space, and that simultaneous observations at
a number of stations determines the location of the
reference axis of the star catalogue within the conventional
terrestrial reference frame. Similarly, Ooe & Sasao (1974)
have explicitly shown that astronomical longitude observations, involving timing the meridional passage of zenith
stars, also involves the location of the reference axis of the
star catalogue within the conventional terrestrial reference
frame. Thus, classical astronomic latitude-longitude observations at an individual station determine the orientation in
space of the station’s local vertical (Jeffreys 1959, 1963;
Atkinson 1973, 1975; Ooe & Sasao 1974; Kinoshita et al.
1979; Sasao & Wahr 1981; Capitaine, Williams &
Seidelmann 1985), and simultaneous astronomic latitudelongitude observations at a globally distributed set of
stations determine the orientation in space of some
reference frame attached to the stations. Since the stations
are attached to some conventional terrestrial reference
frame by the values of their nominal latitudes and
longitudes, then classical astronomic latitude-longitude
observations determine the orientation of the t-frame with
respect to the i-frame.
The very long baseline interferometry (VLBI) technique
measures the difference in the arrival time of a radio signal
at two or more radio telescopes that are simultaneously
listening to the same distant source (e.g. Shapiro 1983;
Lambeck 1988, chapter 8). This technique is therefore
sensitive to processes that change the relative position of the
radio telescopes with respect to the source, such as a change
in the orientation of the Earth in space. If just two
telescopes are observing the same sources, then only two
components of the Earth’s orientation in space can be
determined. A change in the orientation of the Earth about
an axis parallel to the baseline vector connecting the two
radio telescopes does not change the relative position of the
telescopes with respect to the source, and hence this
166
R . S. Gross
component of the Earth’s orientation is not determinable
from single baseline VLBI observations. Multibaseline
VLBI observations with satisfactory geometry can determine all three components of the Earth’s orientation with
respect to a reference frame determined by the locations of
the radio sources. Thus, VLBI measurements can determine
(among other things) the orientation of a conventional
terrestrial reference frame (defined by the adopted latitudes
and longitudes of a set of radio telescopes) with respect to
an (herein assumed) inertial, conventional celestial reference frame (defined by the adopted declinations and right
ascensions of a set of distant radio sources).
In the technique of satellite laser ranging (SLR), the
round-trip time of flight of laser light pulses travelling
between a ground-based laser and a retroreflector mounted
on some satellite is accurately measured (e.g. Lambeck
1988, chapter 6). This time-of-flight measurement is
converted into a distance, or range, measurement by using
the speed of light and correcting for such effects as
atmospheric path delay, satellite centre-of-mass offset, etc.
The SLR technique is therefore sensitive to any process that
changes the distance between the retroreflector mounted on
the satellite and the ground-based laser system, such as that
caused by a change in the Earth’s orientation. Thus, SLR
measurements can determine (among other things) the
orientation of a conventional terrestrial reference frame
(defined by the adopted latitudes and longitudes of a set of
laser ranging stations) with respect to a reference frame
determined by the satellite’s orbit. Lunar laser ranging
(LLR) is a technique similar to that of satellite laser ranging
but where the retroreflector is located on the surface of the
Moon (placed there by Apollo astronauts and unmanned
Soviet spacecraft), rather than on some satellite. LLR can
therefore determine the orientation of a conventional
terrestrial reference frame with respect to a reference frame
determined by the Moon’s orbit.
In the reduction of the observing data taken by the
modern, space-geodetic observing techniques (such as
VLBI), the body-fixed coordinates (after accounting for the
solid-Earth and ocean-loading tidal displacements) of some
observing station are transformed to its coordinates in an
inertial (or quasi-inertial) reference frame by applying a
series of rotation matrices. Let r,(t) denote the components
of some position vector as given in some rotating,
body-fixed reference frame (this position vector may be
time-dependent due to, for instance, plate-tectonic motions). Let ri(t) denote the components of this same position
vector but now given in some inertial, space-fixed reference
frame. These two components of the same position vector
are related to each other by a frame transformation, which
can be written as a series of rotation matrices (e.g. Sovers
1991):
ri(t) = PNUXYr,(t).
(19)
where P accounts for the precession of the Earth, N
accounts for the nutational motion of the Earth, U accounts
for the spin of the Earth, and the product XY accounts for
the Earth’s polar motion. Models (or pre-determined
values) of the Earth’s precession, nutation, spin, and polar
motion are used to fill the elements of the P, N, U, X, and Y
matrices when actually applying (19) to transform the
body-fixed coordinates of some observing station to its
space-fixed coordinates. Observations are then fit to
computed values in order to obtain corrections to the a
priori model parameters.
The currently accepted model for the Earth’s precession
and nutations (Seidelmann 1982; McCarthy 1989) applies to
the celestial ephemeris pole, which coincides with the B-axis
of Wahr (1981). The B-axis of Wahr (1981) is an averaged
figure axis of Earth model 1066A (Gilbert & Dziewonski
1975) where the averaging procedure is meant to average
out the effects of the body tides on the figure axis. Thus, the
celestial ephemeris pole has the desired property of not
being displaced in response to the body tides, where the
instantaneous figure axis does move in response to the body
tides.
Following the precession P and nutation N matrices, the
next rotation matrix in the sequence of transformations (19)
is the matrix U which represents a large rotation about some
axis. The axis about which this large rotation is taken is the
celestial ephemeris pole, since this is the axis to which the
precession P and nutation N rotation matrices apply.
Similarly, the product of the transformation matrices X and
Y represents the polar motion of the celestial ephemeris
pole. That is, the elements of the rotation matrices X and Y
involve the location of the celestial ephemeris pole within
the rotating, body-fixed reference frame (e.g. Sovers 1991):
XY=(
cosPMX 0 -sinPMX
0
1
sinPMX 0 cosPMX
X
=(
i‘
0
0 COSPMY sinPMY
0 -sinPMY cosPMY
O
)
z;)
O1
01
(20)
PMX -PMY
1
where PMX and PMY are the polar motion quantities
reported by Earth rotation services (with PMY being
positive towards 90“W longitude), and where series
expansions for the sine and cosine functions have been used
in order to obtain the final matrix, with terms of second
order and higher in the small quantities PMX and PMY
being neglected.
In the above, the celestial ephemeris pole was identified
with the B-axis of Wahr (1981). This was done because the
elements of the nutation matrix N are filled with the results
of a model for the nutational motions of Wahr’s B-axis.
Each of the transformations P, N, U, X, and Y must refer to
the same axis. Thus, if one transformation matrix explicitly
refers to the B-axis, then they must all refer to this same
axis. However, Moritz & Mueller (1988, section 3.6) argue
that the celestial ephemeris pole should be interpreted as an
averaged angular momentum axis of the Earth. For the
purposes of this paper, the physical interpretation of the
celestial ephemeris pole is unimportant. It is merely
sufficient to realize that the celestial ephemeris pole
corresponds to that axis about which the large rotation is
taken in the transformation matrix U . It is this realization
that is exploited below when deriving an expression for the
location of the rotation axis in terms of the location of the
celestial ephemeris pole.
Theory and observations of polar motion
Thus, to summarize this section, modern Earth rotation
services do not report the location of the rotation axis within
some rotating, body-fixed reference frame [that is, Earth
rotation services do not directly report the m i that are
needed in equations (6-8)], but rather report the location of
the celestial ephemeris pole. This does not imply that the
modern, space-geodetic observing techniques are not
sensitive to the location of the Earth’s instantaneous
rotation vector. It is merely an accepted convention that the
location of the celestial ephemeris pole be reported, rather
than the location of some other axis such as the Earth’s
instantaneous rotation axis, or its instantaneous angular
momentum axis, or any other well-defined axis. The
advantage of reporting the location of the celestial
ephemeris pole is that this pole has the desired property of
exhibiting no nearly diurnal motions in either the rotating,
body-fixed reference frame or the inertial, space-fixed
reference frame (Seidelmann 1982; Moritz & Mueller 1988,
chapter 9). In order to relate the theory of the Earth’s
rotation to observations, it is necessary to obtain an
expression for the location (given within the t-frame) of the
rotation axis in terms of reported quantities, namely, in
terms of the location (also given in the t-frame) of the
celestial ephemeris pole (which is just the axis about which
the large rotation is taken in the transformation matrix U).
This can be accomplished by considering the properties of
the time-dependent transformation matrix that relates the
components of a vector in the i-frame to its components in
the t-frame. In this consideration it will be assumed that no
external torques exist, and hence that the Earth model does
not experience luni-solar precession and nutation. This is
equivalent to assuming that the precession and nutational
motions of the Earth are completely understood and can be
perfectly modelled and removed from Earth rotation
observations. The deformable Earth model does, however,
experience wobble and variations in rotation rate. The
derivation given here of the rotation vector in terms of the
coordinate transformation matrix and its time rate-of-change
closely follows that of Chandrasekhar (1969, section 25);
also see Kinoshita et al. (1979) and Moritz & Mueller (1988,
section 2.3.1).
KINEMATIC DESCRIPTION OF T H E
EARTH’S ROTATION
Consider a body-fixed reference frame (the t-frame) rotating
with respect to an inertial, space-fixed reference frame (the
i-frame). Let the origins of these two reference frames
coincide, but make no assumptions about their relative
orientation. In particular, it is not necessary to assume that
the axes of the two reference frames are momentarily
coincident. There exists a transformation matrix A(t) that
relates the components r,(t) of a position vector given in the
inertial frame to its components rt(t) given in the body-fixed
frame
rI(t) = A(r)ri(t).
The transformation matrix is time dependent since the
t-frame is assumed to be rotating with respect to the i-frame,
and hence its orientation in space is continually changing.
Since A(t) represents an orthogonal transformation (e.g.
Goldstein 1950, section 4.2), the inverse of A(?)is given by
167
its transpose:
A(t)AT(t)= I
where the superscript T denotes the transpose and I is the
identity matrix.
From the time derivative of (22):
dA
-AT(?)
dr
dAT
+ A(t) =0
dt
it is clear that the matrix W ( t ) defined by
dA
W(t) -AT(t)
dt
is antisymmetric since combining (23) and (24) yields
W(t) + WT(r) = 0.
(25)
A vector can be associated with any antisymmetric matrix,
and vice versa:
WJ= &*lkWk
(26)
where summation of repeated indices is assumed, and E , , ~is
the Levi-Civita, or alternating, tensor defined to be 0 if any
two of the indices ijk are identical, +1 if ijk is an even
permutation of 123, and -1 if ijk is an odd permutation of
123. The elements W k of the vector o are the three
independent elements of the antisymmetric matrix W.
In order to assess the physical significance of w, consider
the coordinate transformation of the time derivative of some
vector r(t). Equation (21) relates the components ri(t) of r(t)
in the inertial frame to its components rt(t) in the body-fixed
frame. Inverting (21) yields
r,(t) = AT(t)rl(t)
(27)
the time derivative of which is
dri dAT
dr,
rI+AT-.
dt
dr
dt
Multiplying (28) by A(t) and using (21-25) produces
A
d r.
dt
d
= - (Ari)- W(Ari).
dt
By using (26) and remembering that ( w X rJi
= &j,kO,(rt)k
this becomes the more familiar
A
dr. d
= - (Aq) + w X (Ari).
dt dr
The left-hand side of this expression represents the time rate
of change of the vector in the inertial frame, but with its
components being given along coordinate axes that
momentarily coincide with the coordinate axes of the
body-fixed frame. The first term on the right-hand side of
(30) represents the time rate of change of the vector in the
body-fixed frame, and the second term on the right-hand
side represents the change of the vector due to rotation of
the body-fixed frame with respect to the inertial frame (e.g.
Goldstein 1950, section 4.8). Thus the vector o associated
with the antisymmetric tensor W defined by (24) is just the
rotation vector describing the rotation of the body-fixed
frame with respect to the inertial, space-fixed frame. The
magnitude of o is the instantaneous angular velocity of the
168
R. S. Gross
t-frame (with respect to the i-frame) as it instantaneously
rotates about an axis given by the direction of w .
From the definition (24) of W ( t ) , an expression for the
components wi of the rotation vector in terms of the
elements of the coordinate transformation matrix A ( t ) can
be obtained. The derivation given here follows that of Ooe
& Sasao (1974); also see Capitaine (1986a,b) and Brzezinski
& Capitaine (1991). For the Earth rotation problems of
interest in this paper, it is assumed that the rotation of the
Earth departs only slightly from a state of uniform rotation
about some axis. In order to derive expressions for the
elements of A ( t ) under this assumption, it is convenient to
introduce another reference frame, the u-frame, that is
uniformly rotating at the rate 51 about its reference axis
(i.e., the %,-axis). Let the origin of the u-frame coincide
with that of the i-frame and the t-frame, and furthermore,
let the reference axis of the u-frame coincide with that of the
i-frame. That is, let the $-axis of the uniformly rotating
reference frame (about which it is rotating) coincide with
the $-axis of the inertial reference frame. A transformation
matrix U ( t ) exists that relates the components ri(t) of some
position vector given in the inertial frame to its components
ru(t) given in the uniformly rotating frame:
ru(t) = U(t)ri(t)
where (e.g. Goldstein 1950, p. 109)
U(t)=
(
cosQt
-s$Rt
:).
sinQt 0
coi51t
In the absence of any perturbing forces, let the initial
dynamical state of the Earth be that of uniform rotation at
the rate 51 about the reference axis of the body-fixed frame
(i.e., the %,-axis). That is, in the initial state, the t-frame
coincides with the u-frame, both of which are uniformly
rotating at the rate Q about the reference axis of the i-frame
(which, of course, coincides with the reference axes of the
t-frame and the u-frame). Under the action of some (in
general, time-dependent) perturbing force, the t-frame will
begin to rotate (with respect to the i-frame) about some axis
(the rotation axis) that will no longer coincide with either
the &-axis, the %,-axis, or the &-axis. Assume that the
perturbing force is small enough that (at any time) the
departure of the instantaneous rotation vector from its
initial state is small. Thus, since the t-frame and u-frame
initially coincided, assume that they always remain close to
each other under the action of the perturbing force. In this
case, at any time, the coordinate transformation matrix B(t)
relating the components r,(t) of some position vector given
in the uniformly rotating frame to its components r,(t) given
in the body-fixed frame,
(33)
represents an infinitesimal transformation matrix, the
general form of which is (e.g. Goldstein 1950, section 4.7)
r,(t) = B ( M t ) >
B(t) =
1
-P3(t)
-p&)
(
P3W
1
-pzW
PlQ)
Pdt)
1
)
paragraph, a sign convention has been used that is different
from that usually employed.
The significance of the elements pi of the infinitesimal
transformation matrix B(c) can be understood by considering the components in the body-fixed frame of a unit vector
aligned with the reference axis of the uniformly rotating
frame. That is, let r,(t) = (0, 0, l)T. Then by (33) and (34),
r,(t) = [ p l ( t ) ,p z ( t ) , I]*. Since r,(t) is a unit vector (to first
order in the pi),then p l ( t ) , p 2 ( t ) , 1 are the direction cosines
of the reference axis of the u-frame in the body-fixed frame.
Or, since the reference axes of the uniformly rotating and
the inertial frames coincide, p l ( t ) , p 2 ( t ) , 1 are the direction
cosines of the %,-axiswith respect to the coordinate axes of
the body-fixed frame. Thus, p l ( t ) and p 2 ( t ) are simply the
polar motion quantities reported by Earth rotation services
[compare (34) with the transpose of (20)], and, as shown
below, p 3 ( t ) is related to changes in the rotation rate of the
Earth model. Note that in the absence of external torques,
so that the Earth model does not undergo h i - s o l a r
precession or nutation, then the total angular momentum
vector of the Earth is constant in space, and hence its
direction can be chosen to coincide with that of the
reference axis of the inertial frame. Thus, in the absence of
external torques, p l ( t ) and p 2 ( t ) locate the Earth’s total
angular momentum vector in the body-fixed reference
frame. But, more generally, p , ( t ) and p 2 ( t ) locate the
reference axis of the u-frame (and of the i-frame since these
two reference axes coincide) in the t-frame.
In order to determine the components w, of the rotation
vector in the body-fixed reference frame, combining (31)
and (33) yields r,(t) = B(t)U(t)ri(t), so that by (21),
A ( t ) = B ( t ) U ( t ) , and thus by (24), W ( t ) becomes (to first
order in the pi):
dP,+ 51
dt
dt
0
---p251
ddtp l
*,P,Q
dt
\
I
.
(35)
By correspondence with (26), the components of the
rotation vector in the body-fixed reference frame are then
w,(t) = QP,(t) +!j2,
w2@)= QPZ(t)
- dl?
(36)
(37)
d t ) = Q +A,
(38)
where the dot denotes time differentiation, and hence by
(16), the m iin terms of the reported valves pi are
(39)
(34)
where here, for reasons discussed below in the next
Equations (39-41) are the desired expressions giving the
location of the rotation vector in the rotating, body-fixed
Theory and observations of polar motion
reference frame in terms of the location of the celestial
ephemeris pole. In order to interpret (41), note that m3 is
related to the time interval (UT1 - TAI) by
d
m,(t)=-(UTl
dt
- TAI)
(42)
where UT1 represents a time-scale kept by the rotating
Earth, and TAI is a reference time-scale based upon atomic
clocks (e.g. Lambeck 1980, p. 63). By combining (41) and
(42) it is therefore seen that p 3 ( t ) , apart from a constant of
integration, is the angle about the ?,-axis (or the %,-axis
since they coincide) through which the Earth has rotated
during the time interval (UTl - TAI):
p 3 ( t ) = Q(UT1 - TAI).
(43)
Of more interest here are the results given by (39) and
(40).By defining p ( t ) as
P(t) =
d t )
+ ipz(t)
(44)
and using (39) and (40), then (13) becomes
By defining x ( t ) from
it is clear that (45) is satisfied by
p(t)
+ L ~ ( t=)x ( t ) .
(47)
*o
Expression (47) is one of the main results of this paper. It
relates reported values of polar motion to the geophysical
mechanisms causing the Earth’s rotational properties to
change. The reported values p ( t ) are the location within the
body-fixed reference frame of the reference axis of the
uniformly rotating reference frame (which coincides with the
reference axis of the inertial, space-fixed reference frame).
By (14) and (46), x ( t ) is a function of the perturbations to
the inertia tensor and relative angular momentum:
1.61
‘ ( ‘ ) = Q ( c - A ) [Q AZ(t) + Ah(t)].
148)
The X-functions were defined by Barnes et al. (1983) as
being more accurately computable from available meteorological data that are the ap-functions. Here it has been shown
that the X-functions naturally follow from the result of
writing the theoretical equation describing variations of the
Earth’s rotation (13) in terms of reported values.
DISCUSSION
Polar motion has been historically defined as the motion of
the rotation axis with respect to the figure axis as observed
within some rotating, body-fixed reference frame and has
been extensively studied using (13) with v(t) given by (14).
However, the location of the rotation axis is not actually
reported by Earth rotation services, and when the linearized
conservation of angular momentum equation (13) is written
in terms of reported values, the resulting expression is (47)
169
with ~ ( t given
)
by (48). Mathematically, (47) has the same
form as (13), and hence the motion within the t-frame of the
location of the reference axis of the u-frame can be expected
to exhibit all of the characteristics of polar motion. The
right-hand side of (47), x ( t ) , can be viewed as a forcing, or
excitation, function similar to v(t).A periodic variation of
~ ( t forces
)
p ( f ) to undergo an oscillation at that same
period, exactly analogous to the motion of m ( t ) forced by
ap(t). In the absence of any forcing, p ( t ) experiences a free
wobble identical to the motion of the rotation axis known as
the Chandler wobble.
Expression (13) is not valid at high frequencies as it was
derived under a number of long-period assumptions (e.g.
Wahr 1982, 1986). Notably, long-period approximations
were made to the equation of motion, in particular to the
centripetal acceleration terms, and to the response of the
core. Also, the oceans were assumed to remain in
equilibrium during the wobble, which is probably valid at
low frequencies but is unlikely to remain valid at sufficiently
high frequencies. No long-period approximations have been
made here in deriving (39-41 j and these relations between
the location of the rotation axis and that being reported by
Earth rotation services are valid at all frequencies.
However, since (13) is valid only at long periods, then (47)
is also valid only at long periods. Evidence for this
long-period validity of (47) is readily obtained upon its
Fourier transformation, yielding P(w ) = H ( w ) X (o) where
P ( w ) is the Fourier transform of p ( t ) , X ( w ) is the Fourier
transform of ~ ( t ) and
,
H ( w ) = u,(u, - w ) - ’ is known as
the Earth’s transfer function. This transfer function is only
resonant at the Chandler frequency a,, and not at the free
core nutation frequency. In order for any transfer function
to be valid at high, nearly diurnal frequencies, it must
exhibit a resonance at the free core nutation frequency.
The difference between the location of the Earth’s
rotation axis and that reported by Earth rotation services is
only important at high (nearly diurnal) frequencies, Using
(12) and (44)to rewrite the equations (39) and (40) giving
the location of the rotation axis m ( t ) in terms of the
reported values p ( t ) produces
i dp
m ( t ) = p ( t )- -52 d f
(49)
which, upon Fourier transformation, becomes
M ( w ) = (1
+X)P(w)
in the frequency domain, where M ( w ) is the Fourier
transform of m(t). At frequencies w < < Q , the term in
parentheses in (50) becomes nearly 1 and the motion of the
reported axis p ( t ) is nearly identical to that of the rotation
axis m ( f ) . At higher, closer to diurnal, frequencies,
however, the term in parenthesis in (SO) is no longer near 1
and the reported axis can no longer be identified with the
rotation axis.
Expressions (47) and (49) are the main results of this
paper. Expression (49), valid at all frequencies, relates the
location of the rotation axis to that actually reported by
Earth rotation services. Expression (47), valid at long
periods, is the Liouville equation written in terms of
reported values. In the past, results being reported by Earth
110
R . S. Gross
rotation services have been interpreted using expression
(13), with q(t) being given by (46). This differs from the
present result (47) by the absence [in (47)] of the i - t e r m .
Eubanks et al. (1990) and Eubanks (1991) have qualitatively
argued on the grounds of the separability of nutations from
high-frequency wobbles that the i - t e r m should be dropped
when interpreting results of Earth rotation observations.
Quantitative justification for the neglect of the i - t e r m when
interpreting long-period polar motion results reported by
Earth rotation services has been given in this paper.
One of the main results of this paper, expression (47), was
derived under the simplification that no external torques are
acting upon the Earth model. Thus any change in the
rotational motion of the Earth model in space is assumed t o
be due solely t o internal dynamical processes. However, in
the presence of external torques, the Earth model will
experience additional rotational perturbations known as the
luni-solar precession and nutations. Observations are only
able to determine the orientation of a rotating, body-fixed
reference frame in space and cannot determine whether a
change in this orientation is due t o an external torque or due
to some internal dynamical process. A spatially long-period
change in the orientation of the Earth becomes nearly
diurnal when viewed within a rotating, body-fixed frame.
Thus, observationally, the effect of some internal dynamical
process acting a t a nearly diurnal frequency cannot be
separated from a n externally generated nutation. In other
words, in the presence of external torques, the question of
how to separate nearly diurnal wobbles from nutations must
be addressed. It is beyond the scope of the present paper t o
discuss this issue, but it will be addressed in a future paper.
ACKNOWLEDGMENTS
Discussions with J. A. Steppe, and the comments of one of
the referees, T. A. Herring, led t o substantial improvement
of this manuscript. The work described in this paper was
performed at t h e Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National
Aeronautics and Space Administration.
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